We described the countable minimal prime graphs with no infinite clique (Theorem 2 [1]}, p.358). From our description there exists four countable prime graphs such that every infinite prime graph with no infinite clique embeds one of these graphs. All these four graphs are bipartite and only two are incomparability graphs. These are \(P_\infty \)  and \(H\) . In fact, \(P_\infty \)  is the incomparability graph of \(P_2\)  and \(H\)  is the incomparability graph of \(P_1\) . Since a poset \(P\)  is minimal prime if and only if \(comp(P)\)  is minimal prime (Proposition 4 of [1]}) it follows that every infinite prime poset with no infinite antichain embeds one of the posets \(P_1,P_1^*,P_2,P_2^*\) .
Theorem REF  asserts that the ages of \(P_1\) , \(P_2\) , \({Comp}(P_1)\)  and \({Comp}(P_2)\)  are minimal prime. The question to know wether or not the age of a relational structure is minimal prime whenever \(R\)  is minimal prime was considered by Oudrar ([3]} Problème 1, page 99). Her and our results suggest that the answer is positive. But, note that a description of minimal prime bichains or posets is not known; the same question for minimal prime ages is unsolved too.
